High precision, electronic measurement and test require resistive divider networks having high resistive ratio stability. This means that the ratios of the resistor values should remain as stable as possible when the network is subjected to environmental and operational influences such as temperature and voltage changes.
In the past, high precision instruments had to use very expensive, physically large, wire-wound resistors in their divider networks. Film resistor networks, while satisfactory for less accurate instruments, were generally incapable of the required ratio stability unless they were specially selected, which made them very expensive. Ratio stability on the order of 0.5 parts per million (ppm) per degree centigrade (ppm/.degree.C.) for ambient temperature changes and 2 ppm for input voltage changes of 1000 volts are involved here.
Ratio stability is affected by three primary factors:
1. the difference in the temperature coefficient of resistance (TCR) of the resistors making up the divider (also known as TCR tracking);
2. the difference in the voltage coefficient of resistance (VCR) of the resistors (also known as VCR tracking); and
3. the difference in the temperatures of the resistors. In considering the affect of factor 1, TCR is defined as: ##EQU1## where R.sub.2 and R.sub.1 are the resistance values of a single resistor at temperatures t.sub.2 and t.sub.1 respectively. The TCR may be either positive or negative.
The differences in the TCR's of the resistors comprising the network, or the TCR tracking, have a most significant effect on ratio stability. In a two resistor network, if the TCR of both resistors comprising the network are identical, the ratio of the two will remain constant as the ambient temperature changes. If the TCR's of the two resistors are not the same, as is usually the case, the ratio due to TCR effects will change as the ambient temperature changes. While TCR's may be either positive or negative, which means that the resistance may either increase or decrease with increasing temperature, the greater the difference in the TCR's of the two resistors, the greater will be the change in the ratio or the poorer will be the ratio stability.
In considering the affect of factor 2, VCR is defined as: ##EQU2## where: R.sub.2 and R.sub.1 are the resistance values of a single resistor at applied voltages E.sub.2 and E.sub.1 respectively.
The VCR of deposited film resistors is always negative, and for well designed, properly manufactured, thin film resistors, the VCR is generally quite low. For example, thin film resistors made from 100 to 200 ohms per square material, typically have VCR's in the range of 0.001 to 0.01 ppm/volt. Hence, a 10 megohm resistor will decrease in ohmic value by 1 to 10 ppm (10 to 100 ohms) when the voltage applied to it is increased 1,000 VDC (e.g. from 100 V to 1100 V).
When considering resistive divider networks, the voltage change is in proportion to the resistor values. Hence, for dividers with ratios greater than 10 to 1, only the VCR of the higher value resistor is significant.
Time wise, the effect of VCR on the absolute value of a film resistor is essentially instantaneous while the effect of TCR on the absolute value of a film resistor depends on the thermal time constant of the resistor. Typically 90% of the temperature rise is complete in less than one minute. The combined effect of VCR and TCR on the resistor value is called power coefficient of resistance or PCR, and is the algebraic sum of the change in resistance of a resistive element due to its VCR and an increase in applied voltage (always negative) and the change in resistance of the same resistive element due to its TCR and the self heating caused by the same increase in applied voltage (may be either positive or negative). The combined effect (PCR) can cause the resistor value to either increase, decrease, or in rare cases, even remain constant.
In considering the affect of factor 3, the relative temperature of the two resistors depends upon three parameters:
1. the power dissipated per unit area by each resistor; PA1 2. the distance between the two resistors; and PA1 3. the thermal conductivity of the substrate.
Consider first parameter number 1, the power dissipation per unit area. The power dissipated by each resistor in the network is a given and is directly proportional to the ohmic value of each resistor. If the area of the network was infinitely large, the temperature rise of all sections, and hence the temperature difference between sections, would be essentially zero. If the area of the network was infinitely small, the temperature rise would be very high, but because all the resistors occupied the same space, there would be no temperature difference between sections. Obviously both cases are impractical, but serve as theoretical boundaries. Actual networks probably average 1/2".times.1".times..025" thick with the individual resistive elements placed side-by-side on the substrate. The higher the total power dissipated and the greater the difference in power dissipated by the individual resistors, the greater will be the difference in temperature between the resistors.
Skipping parameter 2 for a moment, consider parameter number 3 next--the thermal conductivity of the substrate. The conductivity of most materials in common use today--steatite, glass, alumina, etc. is relatively poor compared to copper. Hence, there will always be a difference in average temperature of high and low dissipative sections.
This leaves only parameter number 2, the distance between the resistors on the substrate. Unfortunately, regardless of how close together the resistors are placed in the conventional side-by-side configuration, there will always be a difference in the average temperature of high and low power dissipative sections; this again coming back to the imperfect thermal conductivity of the substrate.
Hence, even if the TCR's of the two resistors are identical, the ratios will still change when the applied voltage is increased (unless the TCR of each resistor is zero, which is virtually impossible). Further, regardless of the method used to deposit the resistive material on the substrate, there is always some random variation in the metallurgy of the film. Hence, the TCR of the resulting metal film, from one edge of the substrate to the opposite edge tends to vary smoothly, although not necessarily linearly, with distance from the reference edge. Hence, it is virtually impossible to have resistors with identical TCR's.
There has been a long felt need for a film resistor divider network where the TCR difference of the individual resistors approaches zero over the operational temperature range and the temperature difference of the individual resistors approaches zero over the operational voltage range.